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8 APPLICATION: THE NUMBER e 171

7. Let I = [0,1 [ and let E be the vector space of regulated complex functions defined in I,
bounded and continuous on the right (i.e. /(*+) =/(0 for t E I).

JH-l -------

f(t)g(t) dt is a nondegenerate positive hermitian form
-i

(see (8.5.3)). Prove that the prehilbert space E thus defined is not complete (use the fact
that the function equal to sin(l/r) for / > 0, to 0 for t = 0, is not in E).
(b) Define the sequence (f_n) of elements of E in the following way:

(1) /o is the constant 1;

(2) for each integer n > 0, let m be the largest integer such that 2^m ^ n, and let

2k 2k 4- 1

n = 2^m + k; f_n is taken as equal to 2^m/2 for —T < f <-----TT-, to —2^m/2 for

2^m 2^m

2k + 1 2k + 2

2»»+i 2^m4

Prove that in the prehilbert space E, (f_n) is an orthonormal system (the **Haar
orthonormal system ").

(c) For each n ^ 0, let V_n be the supspace of E generated by the f_k of indices k ^ n.
Show that there is a decomposition of I into n +1 intervals of type [a, /3[ without
common points, such that, in each of these intervals, every function belonging to V_n is
constant; conversely, every function having that property belongs to V_n (consider the
dimension of the vector subspace of E generated by these functions).

(d) Let g be an arbitrary function of E, h its orthogonal projection (Section 6.3) on V_n;
show that in each of the intervals [a, j3[ in which all the functions of V_n are constant,

"*'£-«

(e) Show, by using (d), that for any function g e E which is continuous in I, the series
of general term (g \ fyf_n(t) is uniformly convergent in I and that its sum is equal 1
Conclude from that result that (f_n) is a total orthonormal system in E.

8. Let/be a regulated real valued function in a compact interval I = [a, b]; let \f(t)\ dt =

c. Show that for any e > 0, there is a real valued continuous function g in I, such that

f^fr\g(t)\ ^ 1 in I, and that f(t)g(t) dt ^ c — e. (Reduce the problem to the case in which/

is a step function.)

8. APPLICATION: THE NUMBER e

For any number a > 0, the function x ->• a* is continuous in R (Section
4.3), hence the function g(x) = f*a* dt is defined and differentiable in R, with

/•jc+l px rx+l

g'(x) = a^x everywhere. Now we have g(x + 1) = cfdt—\a^ldt + \ a¹ dt.

JO JO J x

But by (8.7.4), f^a' dt = ^ a^x+u du = a^x ^ a" du; as a*^inf(a, 1) for

Jx Jo JO

0 < x ^ 1, c = f a¹⁴ du is > 0 by (8.5.3), hence we can write
Jo



	8 APPLICATION: THE NUMBER e 171 7. Let I = [0,1 [ and let E be the vector space of regulated complex functions defined in I, bounded and continuous on the right (i.e. /(+) =/(0 for t E* I). JH-l ------- f(t)g(t) dt is a nondegenerate positive hermitian form -i (see (8.5.3)). Prove that the prehilbert space E thus defined is not complete (use the fact that the function equal to sin(l/r) for / > 0, to 0 for t = 0, is not in E). (b) Define the sequence (f_n) of elements of E in the following way: (1) /o is the constant 1; (2) for each integer n > 0, let m be the largest integer such that 2^m ^ n, and let 2k 2k 4- 1 n = 2^m + k; f_n is taken as equal to 2^m/2 for —T < f <-----TT-, to —2^m/2 for 2^m 2^m 2k + 1 2k + 2

	2»»+i 2^m4 Prove that in the prehilbert space E, (f_n) is an orthonormal system (the *Haar orthonormal system "). (c) For each n ^ 0, let V_n be the supspace of E generated by the f_k* of indices k ^ n. Show that there is a decomposition of I into n +1 intervals of type [a, /3[ without common points, such that, in each of these intervals, every function belonging to V_n is constant; conversely, every function having that property belongs to V_n (consider the dimension of the vector subspace of E generated by these functions). (d) Let g be an arbitrary function of E, h its orthogonal projection (Section 6.3) on V_n; show that in each of the intervals [a, j3[ in which all the functions of V_n are constant, 1

	"'£-« (e) Show, by using (d), that for any function g e* E which is continuous in I, the series of general term (g \ fyf_n(t) is uniformly convergent in I and that its sum is equal 1 Conclude from that result that (f_n) is a total orthonormal system in E.

	8. Let/be a regulated real valued function in a compact interval I = [a, b]; let \f(t)\ dt = Ja c. Show that for any e > 0, there is a real valued continuous function g in I, such that f^fr\g(t)\ ^ 1 in I, and that f(t)g(t) dt ^ c — e. (Reduce the problem to the case in which/ Ja is a step function.)

	8. APPLICATION: THE NUMBER e

	For any number a > 0, the function x ->• a* is continuous in R (Section 4.3), hence the function g(x) = fa dt is defined and differentiable in R, with /•jc+l px rx+l g'(x) = a^x everywhere. Now we have g(x + 1) = cfdt—\a^ldt + \ a¹ dt. JO JO J x But by (8.7.4), f^a' dt = ^ a^x+u du = a^x ^ a" du; as a^inf(a,* 1) for Jx Jo JO 0 < x ^ 1, c = f a¹⁴ du is > 0 by (8.5.3), hence we can write Jo